The X-ray CT apparatus is an apparatus that obtains a tomographic image of an object by irradiating the object with fan-beam shaped X-rays or cone-beam shaped X-rays (conical or pyramid beam shaped X-rays), measuring X-rays transmitted through the object using an X-ray detector, and reconstructing the measurement data from multiple directions.
Image reconstruction methods in the X-ray CT apparatus are largely divided into an analysis method and an iterative approximation method. The analysis method is a method of solving a problem analytically on the basis of a projection cutting plane theorem. The iterative approximation method is a method of mathematically modeling an observation system that has acquired the projection data and estimating the best image with a repetition method on the basis of the mathematical model.
When both methods are compared, the advantage of the analysis method is that the amount of computation is overwhelmingly small since a reconstructed image is directly obtained from the actual projection data. On the other hand, the advantage of the iterative approximation method is that the quantum noise on the image or artifacts (such as cone beam artifacts) generated in the analysis method can be reduced since the physical process up to the acquisition of projection data and statistical fluctuations included in the actual projection data can be considered as a mathematical model and a statistical model, respectively.
Conventionally, as an image reconstruction method in multi-slice CT, the Feldkamp method that is an analysis method or an improved method of the Feldkamp method has been mainly used due to the small amount of computation. However, practical applications of the iterative approximation method are also beginning to be considered with the development of high-performance computers in recent years.
The iterative approximation method is a method of setting the evaluation index of an image in advance and updating the image iteratively so that the evaluation value obtained by quantifying the evaluation index takes a maximum or minimum value. As the evaluation index, discrepancy between actual projection data and forward projection data obtained by converting an image into projection data in the update process, stochastic plausibility, or the like is used. A function for calculating the evaluation value is called an evaluation function.
An iterative approximation method using a penalized weighted least-square error function as an evaluation function has been proposed in NPL 1. In the methods proposed up to now, matrices that are transposed matrices of each other in forward projection processing and back projection processing are generally applied as proposed in NPL 1.
On the other hand, although there are few methods, iterative approximation methods to apply matrices that are not transposed matrices of each other in forward projection processing and back projection processing have also been proposed. A method of performing iterative update by applying a view-direction weight to the back projection processing has been proposed in NPL 2.
Hereinafter, the back projection processing in which the view-direction weight is used is called “view-direction weighted back projection processing”.
The view-direction weighted back projection processing itself has been proposed in NPL 3, and is a technique used in the analysis method. The view-direction weighted back projection processing has the following advantages.                (1) Redundancy of projection data can be eliminated.        (2) Time resolution can be improved.        
When matrices that are not transposed matrices of each other in the forward projection processing and the back projection processing are applied, a relaxation coefficient related to the speed and stability of convergence in the iterative approximation method is included in the update expression for iterative update. It is necessary to set the relaxation coefficient in a specific range for stable convergence in the iterative approximation method.
NPL 2 discloses that the relaxation coefficient is determined experientially. On the other hand, a method of calculating the relaxation coefficient using a power law has been proposed in NPL 4. The power law is an iterative solution technique to calculate the maximum eigenvalue of a certain matrix.
Meanwhile, when a spiral scan is performed at high bed movement speed, actual projection data in a row direction is insufficient. Accordingly, there is a problem in that the region where an image is obtained by the iterative approximation method is restricted.
In order to solve this problem, PTL 1 discloses a method of generating virtual row data and virtual channel data by extending actual projection data and then performing back projection processing in the analysis method. If the method disclosed in PTL 1 can be applied to the iterative approximation method, it is possible to relax the restriction of the region.
Hereinafter, the processing of generating virtual row data and virtual channel data by extending actual projection data as in the method disclosed in PTL 1 is called “data extension type back projection processing”.